Rigorous a priori error bounds for floating-point computations are derived. We will show that using interval tools in combination with function and operator overloading such bounds can be computed on a computer automatically in a very convenient way. The bounds are of worst case type. They hold uniformly for the specified domain of input values. That means, whenever the floating point computation is repeated later on with any set of point input values from that domain the difference of the exact result and the computed result is guaranteed to be smaller than the a priori error bound. Our techniques can be used to get reliable a priori error bounds for already existing program code. Here, loops, recursion, and iterations are allowed. To demonstrate the power of the methods several examples are given.